By Christian Pech
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Span(v1 , . . , vn ) = Ul = V. Then the matrix representation of X with respect to this basis is of the shape: X1,1 X1,2 . . X1,l 0 X2,2 . . X2,l .. .. . . . 0 0 . . Xl,l where Xi,i is the matrix representation of X on Ui /Ui−1 . From this, it follows that l χV (X) = l tr(Xi,i ) = i=1 χUi /Ui−1 (X). i=1 The radical of A is defined to be the set of all elements of A that are in the kernel of every irreducible representation of A. Algebras with trivial radical are called semisimple.